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Chapter 2: Functions and Graphs Please review this lecture (from MATH 1100 class) before you begin the section 5.7 (Inverse Trigonometric functions) 2.6 Combinations of Functions; Composite Functions 2.7 Inverse Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Find the domain of a function. Combine functions using the algebra of functions, specifying domains. Form composite functions. Determine domains for composite functions. Write functions as compositions.

Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in a square root of a negative number.

Example: Finding the Domain of a Function
Find the domain of the function Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero. We exclude 7 and – 7 from the domain of g. The domain of g is

The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum f + g, the difference, f – g, the product fg, and the quotient are functions whose domains are the set of all real numbers common to the domains of f and defined as follows: 1. Sum: 2. Difference: 3. Product: 4. Quotient:

Example: Combining Functions
Let and Find each of the following: a. b. The domain of The domain of f(x) has no restrictions. The domain of g(x) has no restrictions. The domain of is

The Composition of Functions
The composition of the function f with g is denoted and is defined by the equation The domain of the composite function is the set of all x such that 1. x is in the domain of g and 2. g(x) is in the domain of f.

Example: Forming Composite Functions
Given and find

Excluding Values from the Domain of
The following values must be excluded from the input x: If x is not in the domain of g, it must not be in the domain of Any x for which g(x) is not in the domain of f must not be in the domain of

Example: Forming a Composite Function and Finding Its Domain
Given and Find

Example: Forming a Composite Function and Finding Its Domain
Given and Find the domain of For g(x), For The domain of is

Example: Writing a Function as a Composition
Express h(x) as a composition of two functions: If and then

Objectives: Verify inverse functions. Find the inverse of a function. Use the horizontal line test to determine if a function has an inverse function. Use the graph of a one-to-one function to graph its inverse function. Find the inverse of a function and graph both functions on the same axes.

Definition of the Inverse of a Function
Let f and g be two functions such that f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f The function g is the inverse of the function f and is denoted f –1 (read “f-inverse). Thus, f(f –1 (x)) = x and f –1(f(x))=x. The domain of f is equal to the range of f –1, and vice versa.

Example: Verifying Inverse Functions
Show that each function is the inverse of the other: and verifies that f and g are inverse functions.

Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows: 1. Replace f(x) with y in the equation for f(x). 2. Interchange x and y. 3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.

Finding the Inverse of a Function (continued)
The equation for the inverse of a function f can be found as follows: 4. If f has an inverse function, replace y in step 3 by f –1(x). We can verify our result by showing that f(f –1 (x)) = x and f –1 (f(x)) = x

Example: Finding the Inverse of a Function
Find the inverse of Step 1 Replace f(x) with y: Step 2 Interchange x and y: Step 3 Solve for y: Step 4 Replace y with f –1 (x):

The Horizontal Line Test for Inverse Functions
A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.

Example: Applying the Horizontal Line Test
Which of the following graphs represent functions that have inverse functions? a. b. Graph b represents a function that has an inverse.

Graphs of f and f – 1 The graph of f –1 is a reflection of the graph of f about the line y = x.

Example: Graphing the Inverse Function
Use the graph of f to draw the graph of f –1

Example: Graphing the Inverse Function (continued)
We verify our solution by observing the reflection of the graph about the line y = x.